Electromagnetic flowmeters

ABSTRACT

An electromagnetic flowmeter with conventional orthogonal electrode and magnetic axes (x, y) intersecting the flowtube axis (z) has its electrodes constituted by two pairs (1, 2 and 3, 4) of electrodes mounted in insulating tubing (5) on each side of the flow passage through the tubing. Four conductors extend respectively from the four electrodes to enable the potential difference between the two electrodes on each side due to an axial component on each side of eddy current electric field in the liquid to be determined for self-calibration purposes in compensating automatically for any extraneous change in the magnetic field and the average potential of each pair of electrodes can also be determined to derive the velocity of flow of the liquid. One of the two pairs of electrodes can be replaced by a single electrode. The electrodes can be point electrodes or electrodes of substantial area and methods are described for mathematically adapting various types of electrodes of the required purpose and to eddy currents in parts of the assembly. The invention can also be applied to an eddy current probe for giving readings of sensitivity at a conventional flowmeter.

This is a divisional of application Ser. No. 07/838,446, filed Mar. 31,1992, now U.S. Pat. No. 5,398,553.

This invention relates to electromagnetic flowmeters of the kind inwhich a liquid having an electrolytic property is caused to flow througha tube on a z axis intersected by an alternating or pulsating magneticfield on a y axis so that a potential difference between electrodes canbe measured for the velocity of flow to be estimated according to theFaraday effect, the electrodes being mounted on opposite sides of thetube on an x axis perpendicular to the y and z axes. Changes in thestrength or distribution of the magnetic field or changes in the contactimpedance between adjoining pipes and the liquid can result ininaccurate flow velocity measurements.

An effect of the alternating or pulsating magnetic field is to producean eddy current in the liquid, whether flowing or not, which has anelectric field with E_(z) and -E_(z) components respectively adjacentthe two electrodes.

It is, therefore, a main object of the present invention to introduceinto an electromagnetic flowmeter a self-calibration function thatcompensates for any magnetic field changes and for changes in contactimpedance between adjoining pipes and the liquid, which compensation isvirtually insensitive to velocity profile effects in the liquid.

According to one aspect of the invention an electromagnetic flowmeterassembly comprises an insulating tube arranged for an electrolyticliquid to flow therethrough while the velocity of the passing liquid isdetermined, a magnet arranged for producing a periodically changingmagnetic field on an axis (y) substantially perpendicular to the tubeaxis (z) and electrodes on opposite sides of the tube positionedaccording to a transverse axis (x) substantially intersecting the tubeaxis and passing though the magnetic field when produced by the magnet,characterised in that on one side of the tube there are two of saidelectrodes mounted side-by-side, being equidistantly spaced from saidtransverse axis (x) on a line substantially perpendicular thereto, andon the other side of the tube at least one electrode positioned inpredetermined relationship to the said two electrodes, and thatconductors extend individually from the electrodes whereby, in theoperation of the assembly, the potential difference between the twoside-by-side electrodes due to an axial component of an eddy currentelectric field in the liquid can be determined for self-calibrationpurposes in compensating automatically for any extraneous change in saidmagnetic field and the potentials of all the electrodes can also bedetermined to derive the velocity of flow of the liquid.

According to another aspect of the invention an electromagneticflowmeter assembly comprises an insulating tube arranged for anelectrolytic liquid to flow therethrough while the velocity of thepassing liquid is determined, a magnet arranged for producing aperiodically changing field on an axis (y) substantially perpendicularto the tube axis (z) and electrodes on opposite sides of the tubepositioned according to a transverse axis (x) substantially intersectingthe tube axis and passing through the magnetic field when produced bythe magnet, characeterised in that on one side of the tube there are twoof said electrodes mounted side-by-side, being equidistantly spaced formsaid transverse axis (x) on a line substantially perpendicular thereto,and on the other side of the tube at least one electrode positioned inpredetermined relationship to the said two electrodes, and thatconductors extend individually and respectively from said electrodes andfrom a current to voltage converting device responsive to theperiodically changing magnetic field to phase-detecting means forproviding self-calibration in compensating automatically for anyextraneous change in said magnetic field to means whereby the potentialsof all the electrodes can also be determined to derive the velocity offlow of the liquid.

In order that the invention may be clearly understood and readilycarried into effect flowmeters in accordance therewith will now bedescribed, by way of example, with reference to the accompanyingdrawings, in which:

FIG. 1 is a diagrammatic perspective view of one form of flowmeter;

FIGS. 2a and 2b are explanatory graphs;

FIG. 3 is a diagrammatic perspective view of another form of flowmeter;

FIGS. 4a and 4b are a cross section and a longitudinal section of aportion of the flowmeter of FIG. 3;

FIGS. 5 and 6 are explanatory diagrams;

FIG. 7 is an explanatory graph;

FIG. 8 is a diagrammatic longitudinal section of a portion of aflowmeter;

FIGS. 9a and 9b are diagrams showing magnetic fields in flowmeters;

FIGS. 10a and 10b are perspective external views of flowmeters;

FIGS. 11 and 13 are diagrammatic representations of electricalconnections in a flowmeter;

FIGS. 12a and 12b are respectively a cross-section and a longitudinalsection of a flowmeter;

FIGS. 12c and 12d are respectively a cross-section and a longitudinalsection of a flowmeter;

FIGS. 14a and 14b are respectively a cross-section and a longitudinalsection of a flowmeter;

FIGS. 15a and 15b are respectively a cross-section and a longitudinalsection of a flowmeter;

FIGS. 16a, 16b, 17a, and 17b show front and side elevations of anarrangement of two electrodes;

FIG. 18 is a diagram of portions of two juxtaposed electrodes;

FIG. 19 is a diagram of a further arrangement of two juxtaposedelectrodes;

FIGS. 20a and 20b, 21a and 21b, and 22a and 22b respectively show frontand side elevations of three further arrangements of juxtaposedelectrodes;

FIG. 23 is a diagrammatic cross-section showing a further detail;

FIG. 24 is a diagram showing another arrangement of electrodes and theirconnections;

FIG. 25 is a diagrammatic perspective view of yet another form offlowmeter.

FIG. 26 is a further explanatory diagram;

FIG. 27 is a circuit diagram showing electronic connections made to anelectromagnetic flowmeter to enable the flowmeter to be used;

FIG. 28a is a perspective view showing a method of using an eddy currentprobe in an electromagnetic flowmeter; and

FIG. 28b is a vertical section showing a detail of FIG. 28a.

A conventional electromagnetic flowmeter measures the velocity of anelectrolytic liquid flowing in a pipe by sensing the potentialdifference generated by the liquid as it moves through an externalmagnetic field. Thus with reference to FIG. 26 the liquid flows (in thez direction) through a tube of insulating material and a magnetic fieldis present (in the y direction). As the liquid moves it generates avoltage across diametrically opposed electrodes 1 and 2 which aresituated on the x axis on the inside walls of the tube. The potentialdifference U₁ -U₂ is proportional to the average liquid velocity v andis thus a measure of that velocity or of the total volume flowrate. Inthe invention means are introduced for compensating for the effects ofvariation in the magnetic field or of variation in the conductivity ofthe internal surfaces of tubes joined to the ends of the meter. Eitherkind of variation can alter the constant of proportionality between U₁-U₂ and v. The method of compensation shown in FIG. 3 involves the useof four electrodes rather than two. Thus each electrode in FIG. 26 isreplaced by a pair of electrodes one slightly upstream and one slightlydownstream of the position of the original electrode. To obtain thevelocity measurement the mean values of the potentials of each pair areused; the difference ΔU=1/2(U₁ +U₂)-1/2×(U₃ +U₄) being proportional tothe average flow velocity v. The constant K of proportionally connectingΔU and v can be obtained from measurements of the potential differencesε₁ =U₁ -U₂ and ε₂ =U₄ -U₃ which arise because of eddy currents inducedin the liquid by the time variation of the magnetic field. Derived inthis way and in certain circumstances, the constant K automaticallyadjusts itself to the correct value whenever the magnetic field changesor whenever the contact impedance of pipes joined onto the flowmeterends changes. This automatic adjustment of K is a consequence of arather subtle mathematical relation between flow induced voltages andeddy current electric fields. The details of this mathematical aspect ofthe invention are given below.

The idea of a self-calibrating electromagnetic flowmeter using an eddycurrent E field (FIG. 9(a)) measurement is based on the mathematicalrelation ##EQU1## which holds (in any flow tube) between the potential Uinduced by a flat profile of speed v and the z component of the E fieldassociated with eddy currents induced in the liquid. iω represents thecyclic excitation of the magnetic field. Assuming no end effects onvirtual current, (1) holds at every point in the liquid, the factore^(-i)ωt expressing the time dependence of the fields.

In its most simple form the self-calibrating flowmeter consists of threesmall contacting electrodes positioned as shown in FIG. 1. Electrode 1is in its usual position but on the other side we have two electrodes 2and 3. 2 is positioned slightly downstream and 3 slightly upstream ofthe usual electrode position. Let δ be the separation of electrodes 2and 3. In normal operation the flow signal ΔU is derived from thepotential U at the end of electrode wires using the formula ##EQU2##where R(x) means the component of X in phase with the reference(derived, for example, from magnet current). To obtain self-calibrationthe flat profile sensitivity S (i.e. the quantity ΔU that would bemeasured for a flat profile of unit speed) is derived using the formula##EQU3## where R⊥ (X) means the component of X one right angle (90°) inadvance of the reference phase. Provided δ is small and there isnegligible magnetic flux through loops in electrode wires it followsfrom (1) that (3) is a valid expression for S.

ADAPTATION FOR VARIOUS WAVEFORMS

Self-calibration can work just as well with non a.c. modes of magnetexcitation. For example with `keyed dc` excitation and the electrodeconfiguration of FIG. 1 the B field or flow signal ##EQU4## and the emf

    ε=U.sub.3 -U.sub.2

have the forms shown in FIGS. 1(a) and 2(b). Since at any point in theliquid U and E_(z) are periodic functions of time we have ##EQU5## Foreach Fourier component, (1) holds, i.e. ##EQU6## Hence ##EQU7## orintegrating between the times t₁ and t₂ FIGS. 2a and 2b at whichmeasurements of ΔU are taken. ##EQU8##

In other words the sensitivity is derived by integrating ε over the timeinterval t₁ to t₂.

MAGNETIC END EFFECTS

Self-calibration compensates exactly for any variation in the magneticfield (shape, strength or phase). Note, however, that to compensate forvariations of the B field which violate normal symmetry, four electrodesare needed (i.e. electrode 1 in FIG. 1 has to be replaced by a pair like2 and 3 on the other side) and measured emfs must be compoundedappropriately. Thus with small point electrodes a fully fledgedself-calibrating flowmeter is as illustrated in FIG. 3 and in acexcitation the flow signal is ##EQU9## and the sensitivity is ##EQU10##

NON-FLAT VELOCITY PROFILES

If an electromagnetic flowmeter is to be self-calibrating in conditionsof variable velocity profile, it is necessary that (i) the flow tube bedesigned to approach an ideal configuration; i.e. magnetic field andvirtual current shapes optimised and (ii) the variations in velocityprofile be not too great. This is seen mathematically as follows: Letthe design configuration have magnetic flux density B_(o) and virtualcurrent j_(o). In production (or in ageing or with end effects . . .etc) B and j will differ from the designed fields B_(o) and j_(o).Therefore suppose B is in fact B_(o) +B' and j is in fact j j_(o) +j'.Let the velocity profile be v_(o) +v' where v_(o) is a flat profile andv' a flow of zero total flow rate. The change in flow signal is now

    δU=∫(v.sub.o +v')·(B.sub.o +B')×(j.sub.o j')dV-∫v.sub.o ·B.sub.o ×j.sub.o dv   (6)

Assuming B'<<B_(o), j'<<j_(o) and v'<<v_(o), then to first order

δU=∫v_(o) ·B_(o) ×j'dV+∫v_(o) ·B'×j_(o) dV+∫v'B_(o) ×j_(o) dV (7)

With self-calibration the estimated mean velocity v is ##EQU11## whereΔU is the flow signal and S the sensitivity obtained by eddy current Efield measurement. Therefore the fractional error δv/v in mean velocityis: ##EQU12## where δU/U and δS/S are the fractional changes in U and S.Now δU is made up of three parts, i.e. the three integrals in (7). Sinceself-calibration compensates exactly for any magnetic field change, thesecond integral causes a change δU/U which is exactly cancelled by thecorresponding change δS/S. Therefore the second integral in (7) causesno error in v. On the other hand the third integral in (7) causes a δU/Uwhich has no counterpart in δS/S (the eddy current E field does notdepend on v ). Therefore the velocity profile error goes straightthrough to an equal fractional error in v. This is the reason that theelectrode/magnet configuration must be optimised. ∇X (B_(o) ×j_(o)) isthen small (in some sense) and the third integral in (7) negligible.

If deviations from a flat profile are not small (e.g. very near a bendor in laminar flow) the higher order terms

    ∫v'·B'×j.sub.o dV+∫v'·B.sub.o ×j'dV(10)

need to be included in (7). Since B' or j' arise from unpredictablecauses ∇×(B'×j_(o)) or ∇×(B_(o) ×j') are not small and errors in v willbe present. To estimate the error we could say that the integrals in(10) have the approximate values v'·B'·j_(o) ·V and v'·B_(o) j'·V wherethe bar here denotes the mean values in the effective volume V of theflowmeter. However we know from experience that purely circulating flowslike v' are, in practice, unlikely to produce flow signals in excess of3% of the magnitude estimates v'·B'·j_(o) ·V and v'·B_(o) j'·V. Thisfact together with the assumption that B' and j' are <<B_(o) and j_(o)respectively means that the higher order terms (10) are probablynegligible even if v' is not small compared with v_(o).

The remaining source of error is the first integral in (7). This arisesfrom changes in virtual current which could be due to end effects,inaccurate positioning of electrodes . . . etc. However changes δU/U dueto end effects on virtual current can be cancelled to a greater orlesser extent by corresponding changes δS/S in sensitivity measurement.

ADAPTATION FOR ELECTRICAL END EFFECTS

Self-calibration does not compensate exactly for end effects on virtualcurrent. This rules out the possibility of very short flow tubesinsensitive to the properties of adjoining pipes. However, partialcompensation to end effects on virtual current can be obtained and thisimplies the possibility of some degree of shortening of flowmeterswithout loss of accuracy.

FIGS. 3, 4 and 12 show embodiments of the invention in which the pairsof electrodes 1, 2 and 3, 4 are held in an annular liner consisting of amass of dielectric material 5 in an expansion 6 of the flowtube 7, themass 5 being formed with a bore 8 in exact register with the internalsurface of the unexpanded portion of the flowtube 7. The expansion 6 maybe regarded as a grounded screen.

For the purpose of analysis the magnet coil, electrical steel, stainlesssteel tube, adjoining tubes (when these are steel, copper or anothermetal) and any other metal parts of the flowtube (except the electrodes,electrode wires, screens . . . etc) are considered to constitute the`magnet`. The rest of space is then termed the `external` region. Thusin an assembly such as shown in FIGS. 3 and 4 the external regionconsists of the liner, the liquid, the electrodes, the electrode wiresextending from the electrode beyond the flowtube, the screens aroundelectrode wires . . . etc. Eddy currents may be present in parts of themagnet and these contribute to the B field in the external region.However, these eddy currents (being in highly conducting material) areindependent of the properties of parts of the external region (e.g.liquid conductivity or contact impedance between liquid and adjoiningpipes) so that in the external region, B depends only on properties ofthe magnet and so is a given external field. Also the E field in partsof the magnet is (to first order) independent of the properties of theexternal region and so provides a definite term in the boundaryconditions for the E, D and j fields in the external region where D=εE.In parts of the magnet we denote the E field by E'. Eddy currents in theexternal region (e.g. in the liquid or electrodes ) are, of course,assumed to produce negligible secondary magnetic fields.

Equations for the fields in the external region can be derived fromMaxwell's equations in their vector form. Effects of previous history(e.g. previous currents across the interface between liquid andadjoining pipes) are neglected and the contact impedance at a metalliquid interface is regarded as a definite complex function of frequencyand position on the interface, and secondary magnetic fields arenegligible. Then the fields are separated into eddy current and flowinduced parts.

The equations for the flow induced potential U in the liquid are (forflat profile flow)

    ∇.sup.2 U=0                                       (11) ##EQU13## These equations determine U in the liquid uniquely (where τ is the contact impedance between liquid and adjoining pipe and σ is the conductivity of the liquid).

In the case of the eddy current E_(z) field in the liquid, the equationsare ##EQU14##

Equations (14), (15) and (16) are similar to equations (11), (12) and(13) and if only the R.H.S. of (16) was zero we would have the identity##EQU15## as required for self-calibration. The condition for effectivecompensation of virtual current end effects is therefore that ##EQU16##is small. Since E_(z) ' is independent of τ we require the separateconditions that E_(z) ' and σE_(r) δτ/δz are both small compared withthe value of the LHS of (16) at points in the liquid in the end regionsof the flow tube. In order of magnitude, the ratio of the two terms onthe LHS of (16) is τσ/b (where b is the tube radius). This quantity cantake any value in practice from a value <<1 to a value >>1. Therefore tocater for all likely situations we require the conditions ##EQU17##

The LHSs of these conditions are evaluated at the wall of the adjoiningtube and the RHSs are orders of magnitude at points in the liquid in theend regions of the flow tube. The second condition is equivalent to##EQU18## lτ is the characteristic distance of variation of τ (18) and(19) are necessary and sufficient conditions for good compensation ofvirtual current end effects under all conditions likely to be met inpractice.

In the least favorable case from the point of view of the effect ofE_(z) ' (i.e. for τ σ/b <<1) the extent to which (18) is satisfied isdirectly the extent to which compensation is achieved. Thus if E_(z)'=1/10th of E_(z) in the end regions of the flowmeter we expectcompensation to within about 10%. In the least favourable case from thepoint of view of the effect of δτ/δz (i.e. for τσ/b >>1) the extent towhich (19) is satisfied is not so directly related to the degree ofcompensation obtained. This is because δτ/δz is probably a randomfunction of position on the liquid/adjoining pipe interface. It isunlikely that this function will be the most unfavourable function sothe degree of compensation achieved is likely to be better than theratio l τ/b might suggest. The effect of the constant (mean) part of τis (in the absence of E_(z) ') exactly compensated by self-calibrationand exact compensation is still present if τ is a function of θ only.

A key requirement for good compensation of virtual current end effectstherefore seems to be that the eddy current E_(z) ' field in theadjoining pipes be small compared with the eddy current E_(z) field inthe liquid at the flow tube ends. This leads us to the calculation ofthe E_(z) fields.

To start off we calculate E_(z) in the liquid with the idealised magnetas illustrated in FIGS. 4a and 4b. The liner on r=b is assumed to extendindefinitely in the ±z directions. Although this configuration does notoccur in practice it serves to obtain the order of magnitude of E_(z) inthe liquid at the ends of a typical flow tube.

For r<f FIGS. 4a and 4b the magnetic potential F(r,θ,z) is ##EQU19##

In (20) A_(m) is a function depending on f, h, α₁, α₂ (see FIGS. 4a and4b).

Then from (20), (14), (15) it is possible to obtain an expression forthe E_(z) field in the liquid which is approximately valid for ξ>h (FIG.8) and perfectly correct for ξ→∞.

In order to calculate E_(z) ' in an adjoining pipe, it is necessary toconsider the interaction of the pipe with the magnetic field justoutside the flow tube.

Various physically different cases arise in practice. In one case thepermeability of adjoining pipe may be low (as in stainless steel) andthe magnetic field may not be greatly affected. The eddy currents andthe E field can in this case be calculated by regarding the magneticfield as a given external field (secondary magnetic fields due to theeddy currents being assumed small). In another case the permeability ofadjoining pipe may be high (as in mild steel or cast steel pipes). Themagnetic field in this case is greatly affected and the eddy currentsand the E field may be effectively confined to a narrow layer at thesurface of the pipe (skin depth).

Although the physics is different the order of magnitude of E_(z) ' inthe various cases is the same. This is essentially because Faraday's Law

     E·dl=-Φ

holds universally and the flux Φ is the same order of magnitude in allcases. By choosing an appropriate contour we obtain in order ofmagnitude

    E.sub.z '=1ωbB                                       (21)

where b is the pipe radius and B the order of magnitude of the magneticflux density at the end of the flow tube.

However, to be more exact in the estimation of E_(z) ' we study in moredetail the nature of the interaction of a permeable conductor with analternating magnetic field and then work out an exact expression for theE field in a pipe situated in the far field of the idealised magnet(assuming secondary magnetic fields are small). The interaction of apermeable conductor with an ac magnetic field is relevant to the problemof predicting magnetic end effects and is therefore worthy ofconsideration from this point of view also.

Of first importance here is the concept of skin depth δ_(s) defined as##EQU20## where μ and δ are the permeability and conductivity of theconductor and ω(=2πf_(o)) the angular frequency of the field. δ_(s) is ameasure of the depth to which the magnetic field can penetrate theconductor. In the present application f_(o) can be 60 Hz (mainsexcitation). In keyed dc excitation or triangular wave form excitationthe fields can be analysed into harmonics ranging from f_(o) =3 H_(z) toabout f_(o) =100 H_(z). Some examples of the magnitude of δ_(s) at 20°C. are given in Table 1.

                  TABLE 1    ______________________________________     ##STR1##            ##STR2##                      ##STR3##                                ##STR4##                                          ##STR5##    ______________________________________    Mild   .sup.- 10.sup.7                     800        3        3.2    steel            (B = .1T)  60       0.7    or                         100       0.6    Cast steel    Stainless           .sup.- 1.4 × 10.sup.6                     1 to 10    3        246 to 76    Steels                      60       55 to 17                               100       43 to 13    Copper .sup. 5.9 × 10.sup.6                     1          3        12                                60       2.7                               100       2.1    ______________________________________

These suggest that when the magnet field of an electromagnetic flowmeterinteracts with adjoining steel pipe (other than stainless) σ_(s) isquite small and the field is hardly able to pass through the thicknessof the pipe except perhaps at the lower frequencies.

By using the basic equations governing the interaction of a permeableconductor with an alternating magnetic field it can be shown withreference to FIGS. 5 and 6 that if the simple field B_(x) =δA/δY,By=-δA/δα (where A=dαχ sin αy) interacts with the permeable conductor inχ<0 then on x=0 ##EQU21## where is short for (α² +i μδω)^(1/2) and α⁻¹is a measure of the spacial dimension of the magnetic field. This givesthe conditions under which simple boundary conditions on B may beassumed on the outside surface of the conductor. If (B)_(t) and (B)_(n)are the tangential and normal components of the magnetic field we canassume

    (B).sub.t =0 1f R.sub.1 >>1                                (24)

or

    (B).sub.n =0 1f R.sub.1 <<1                                (25)

Case (24) occurs when ω→0 (provided μ/μ_(o) >>1) and represents theusual boundary condition at a highly permeable material. Case (25)occurs when ω→∞ (for any μ/μ_(o)) and is the case of total exclusion ofthe field from the conductor. Note that δ_(s) may be <<1/α (thecharacteristic dimension of the B field) and yet (24) may still hold forsufficiently large μ/μ_(o). Values of R₁ for mild steel or cast steeland for various 1/α values and frequencies are given in table 2.

                  TABLE 2    ______________________________________                1/α (mm)                10   50        100     150    ______________________________________    f.sub.o (H.sub.%)             3        180    35      18    12             60       40     8       4     27            100       33     6.4     3.3   2.1    ______________________________________

These are obtained using (23) which, since (α δ_(s))⁴ <<1 in this case,reduces to R₁ =800 αδ_(s) /√2. Table 2 suggests that when the magneticfield of electromagnetic flowmeters interacts with adjoining steel pipes(other than stainless) boundary condition (24) operates at leastapproximately except at the upper limits of flowtube sizes andfrequencies.

It can be shown with respect to FIG. 6 that for the x and y componentsof flux density in the material, the following expressions exist##EQU22## When skin depth is not small (i.e. δ_(s) ˜1/α) flux density isof course of the same order of magnitude in the material as outside it.When

    δ.sub.s <<1/α                                  (27)

flux is `returning` in a narrow layer of the material and so fluxdensity can be larger in the material than outside it (see FIG. 6). Thisonly happens, however, when μ/μ_(o) is sufficiently high to ensure thatB_(x) is not <<B_(y) on x=0+. The order of magnitude of the externalflux density in the vicinity of the material is α. The material does notchange this order of magnitude. From (26) the order of magnitude ofB_(y) in the material is ##EQU23##

Therefore the ratio of relative values of flux density in and out of thematerial is of the order ##EQU24##

This is large when δ_(s) <<1/α and μ>>μ_(o).

The result (28) is important in the flowmeter problem for determiningthe correct value to use for μ. The B versus H characteristic inmaterials containing iron is not linear. For mild steel and cast steel,for example, we have the characteristics shown in FIG. 7. The `correct`value for μ in these orders of magnitude estimates therefore depends onthe flux density in the material. In Tables 1 and 2 we have assumedμ/μ_(o) =800. This amounts to assuming that flux density in mild or caststeel is somewhere around 0.1 T. To see that this assumption is validconsider a 100 mm (4") diameter flow tube. The flux density in the mainfield is ˜0.01 T.

The flux density at the ends of the flow tube will be much less; 0.001 Tat most. Using (28) with 1/α=50 mm. δ_(s) =1 mm and μ/μ_(ob) =800 wefind |B_(in) |˜94×0.001=0.094 T in agreement with the originalassumption. Since, in (28) the term μ_(o) /μ is negligible, similarremarks hold for all other flow tube sizes.

The E field is given by E=iωA+∇φ where B=∇×A. In the 2-D problemconsidered here we have (A)_(x) =0. On the inside surface of the tubeδφ/δx=0 on x=0- so by ∇² φ0, φ=0 for x<0. Thus E=iωA+∇φ gives

    E.sub.z =1ωA

and so it can be shown that ##EQU25##

It is of some interest to see how this compares with the estimate (21).First let B be the order of magnitude of the external field in thevicinity of the body. This is α, therefore for an external field B (29)gives ##EQU26##

Now 1/α is the characteristic dimension of the B field and is equivalentto b in (21). Using (23) the factor ##EQU27## can differ from 1 onlywhen R₁ <<1. F is then <<1. This can happen, when, for example, ω→∞ or1/α→∞ and corresponds to the situation where, by (23), the normalcomponent of B at the surface of the conductor is << the tangentialcomponent. Physically B is excluded from the conductor so that theinduced E field (in the conductor falls.

From Table 2 we see that in the electromagnetic flowmeter situation, Fis generally of order 1 for adjoining steel pipes other than stainless.For stainless steel or copper pipes R can be <<1 especially in thelarger flow tubes and at the higher frequencies. The estimate (21) istherefore a worst case, i.e. it may overstate the strength of E_(z) ' inthe wall in certain cases.

We consider the worst case as far as adjoining tube material, flow tubesize and frequency are concerned. That is, we assume the secondarymagnetic field due to eddy currents in adjoining pipe is << the primaryfield. Such is the case, for example, with stainless steel adjoiningpipes in a 100 mm flowtube working at 60 Hz. To make a fair comparisonwith the E_(z) field calculated in relation to (20) we assume theidealised magnet as used in that section. The problem is therefore tofind E_(z) ' in a conducting pipe situated in the field of FIGS. 4a and4b (see FIG. 8). Note we neglect the effect of flanges.

Since the pipe wall thickness is <<b we can treat it as infinitely thinwithout altering the conditions of the problem. We use the far field ofthe magnet for which the magnetic potential is a simplified function F(γ, θ, ξ).

By using the equations for the E field in the pipe including the term ∇φsuch that ∇² φ=0 it is possible to obtain an equation giving E_(z) '.Hence an expression is obtained for the ration R₂ of the maximum valuesof the z component of electric fields in the adjoining pipe and in theliquid near the ends of the meter. Thus ##EQU28##

Referring to the values b, d, f, h as represented in FIG. 8, forrealistic values of f/b, d/b and h/f, R₂ is quite small (see Tables 3, 4and 5). This is encouraging for the generality of the self-calibrationtechnique in that the first of the conditions (18) and (19) required forgood compensation of virtual current end effects looks likely to besatisfied in practice.

                  TABLE 3    ______________________________________     ##STR6##              f/b              1     1.2     1.3     1.4   1.5    ______________________________________     ##STR7##            .75 1   0.63 0.39                            .095 .067                                  .094 .072                                        .085 .068                                              .071 .059            1.25    .024    .048  .054  .055  .049            1.5     .015    .034  .041  .044  .041            1.75    .009    .024  .031  .035  .035    ______________________________________

                  TABLE 4    ______________________________________     ##STR8##              f/b              1     1.2     1.3     1.4   1.5    ______________________________________     ##STR9##            .75 1   .084 .051                            .117 .084                                  .114 .087                                        .101 .081                                              .082 .068            1.25    .031    .060  .066  .065  .057            1.5     .019    .043  .050  .052  .048            1.75    .011    .030  .038  .041  .040    ______________________________________

                  TABLE 5    ______________________________________     ##STR10##              f/b              1     1.2     1.3     1.4   1.5    ______________________________________     ##STR11##            1 1.25  .067 .041                            .105 .075                                  .106 .081                                        .097 .077                                              .080 .067            1.25    .025    .053  .061  .062  .056            1.75    .015    .038  .046  .049  .047    ______________________________________

The physical reason for the small values of R₂ is, it seems, as follows.The E field in the liquid circulates around a loop through which passesflux from the main magnetic field in the flow tube (FIG. 9(a)). The Efield in an adjoining pipe, however, circulates around a loop throughwhich passes flux from the fringe magnetic field (FIG. 9(b)). Thus thecentral magnetic flux contributes to E in the liquid at the ends of flowtube but not to E' in the adjoining pipes.

There is a difference between wafer and flanged flow tubes as far as thelikely value of E' (the electric field in adjoining pipes) is concerned.Consider a closed loop abcdefghijkl in the metal structure consisting ofadjoining pipes, flanges, bolts and (in the flanged tube) the stainlesssteel tube (FIG. 10(a) and (b)). In the wafer design (FIG. 10(a)) themain magnetic flux returns within the loop and so does not contribute tothe total flux through the loop. In the flanged design (FIG. 10(b)),however the main flux returns in the electrical steel which lies outsidethe loop and so the main flux does appear to contribute to the E' fieldin the adjoining pipes. To avoid this flux linkage in the flanged designit is desirable to insulate the flanges of the flow tube from those ofthe adjoining pipes, though the small contacting area and the contactresistance between bolts and flanges may on its own provide sufficientinsulation. The estimate of E_(z) ' which leads to Tables 3, 4, 5 of R₂values does, of course, assume no contribution to E_(z) ' from the mainmagnetic flux.

LARGE AREA NON-CONTACTING ELECTRODES

We now consider the possibility of extending the self-calibrationprinciple to large area contactless electrodes. Contactless electrodesare electrodes separated from the liquid by a thin layer of insulatingmaterial. Each such electrode makes contact with the liquid through thecapacitance formed between itself and the liquid. Contactless electrodesare not exposed to the fluid and therefore cannot be harmed by it. Largearea contactless electrodes result in better averaging over velocityprofiles and give rise to reduced turbulent noise signals.

The basic question of interest now is whether or not a flow tube withfour extended area contactless electrodes (FIG. 11) works in aself-calibrating fashion. If so this allows self-calibration in flowtubes virtually insensitive to velocity profile effects of any kind.

It is assumed that the bias relation (1) (between the flow inducedpotential U and eddy field component E_(z)) holds at every point in theliquid. (1) holds, under certain conditions, even if end effects onvirtual current are present. The question now, however, is whetherrelation (5) holds where U₁, U₂, U₃ and U₄ are the potentials at theends of wires connected to the contactless electrodes in FIG. 11 and δis now some effective separation between adjacent electrodes. For pointelectrodes (5) follows immediately from (1) (given that δ is small)because potentials at points in the liquid are measured. Now, however,the flat profile signal and the eddy current emf between adjacentelectrodes must be recalculated and compared.

The flow signal with contactless electrodes is related to the mean valueof flow induced potential U over the electrodes. For self-calibrationwith contactless electrodes we therefore require that the measured emf εbe related to the same mean value of the eddy current E_(z) field overthe electrodes. It will be shown that this is not always the case butthat with special laminated electrodes the necessary relation can beobtained.

We will consider a single Fourier component of the fields with frequencyω. However, the main conclusions hold true (by Fourier analysis) for anyform of periodic magnetic excitation.

Equations for the flow induced fields are ##EQU29## where E₁ and j₁originate from v×B. Thus

    E.sub.1 =-∇U                                      (33)

In the liquid U is determined independently of the physical propertiesof the rest of the external region by ##EQU30##

At the liquid/liner interface the 4th of (32) gives that U iscontinuous. Therefore to find U in parts of the external region otherthan the liquid U can be regarded as given on the liner/liquidinterface.

For example, if (as shown in FIGS. 12a and 12b) the liquid/linerinterface is e and electrodes 1, 2 and 3, 4 are imbedded in an insulator5 which is surrounded by a grounded screen c, the boundary value problemfor U in the insulator is as shown in FIGS. 12c and 12d, U_(g) being thevalue at the wall of the known potential U in the liquid.

If in FIGS. 12a-12d the spacing δ' is constant and small over the wholearea of each electrode the flow induced potential U on each electrodesettles to the mean value of U_(g) over the electrode area. U_(g) ishere the value at the wall of the known potential U in the liquid. Theflow signal ΔU for a flat profile after phase sensitive detection istherefore ##EQU31## where U_(g12) and U_(g34) are the mean values of Uin the liquid (as determined by (34), (35) and (36)) over the areas ofelectrodes 1, 2 and 3, 4 respectively, where R(X) means the component ofX in phase with the reference.

With reference to FIG. 13 the emf ε across the far ends a and g of wiresabc and efg connected to adjacent electrodes is found by a directapplication of Faraday's Law. We form a closed loop abcdefgha (FIGS. 13and 18) and obtain ##EQU32## where ε is the area of the surface boundedby the loop or rather the part of that surface in the magnetic field.But ##EQU33## and E in the wires is zero (assuming very thin wires),therefore ##EQU34##

Therefore to find ε we need to know the integral of E along some linejoining the junctions e and c on the electrodes and the flux of Bthrough a surface bounded by this line and the wires.

In the liquid δ>>ωε and E' is assumed negligible so E is determined bythe equations ##EQU35## independently of the physical properties of therest of the external region.

At the liquid/liner interface (E)_(t) is continuous. Therefore infinding E in parts of the external region other than the liquid we canregard (E)_(t) as given on the liner/liquid interface. Similarly inother highly conducting parts (e.g. the electrodes) E is determined bythe equations ##EQU36## independently of the properties of the rest ofthe external region. Therefore E can be worked out first in eachconducting part and then in the remainder of the external region usingthe known value of (E)_(t) on the surface of the conductors. Note thatto work out E in the non-conducting part of the external region we needalso information regarding the total current passing in or out of theelectrodes via the electrode wires.

For example, in the configuration shown in FIGS. 12a and 12b theboundary value problem for E in the insulator is as indicated in FIGS.14a and 14b. Zero net current is assumed to pass into each electrode. Asa procedure for solving this problem we can start by finding the fieldthat would be present if the electrodes were absent and then compute theextra field due to the presence of the electrodes.

Let E_(e) denote the E field that would be present in the insulator ifthe electrodes were removed (i.e. replaced by insulator). Let E_(i)denote the E field inside an electrode when in position. As noted aboveE_(i) depends only on the B field and electrode geometry. The totalfield E in the insulator may now be written as

    E=E.sub.e -∇φ                                 (41)

where φ satisfies the equation

∇² φ=0 (42)

and the boundary conditions ##EQU37##

Note that the second of (43) determines φ over an electrode surface onlyto within an additive constant. This constant can be found from theknowledge of the net current passing into the electrode. Thus if, in themeasurement of the potentials at the end of the wires connected to theelectrodes, zero current is drawn we have the extra condition

    ∫(E.sub.e -∇φ)·dε=0

where the integral is conducted over the whole surface of the electrode.Since ∇·E_(e) =0 this reduces to

    ∫∇φ·dε=0                (44)

It is clear that when φ satisfies the above requirements (summarised inFIG. 15) E, as given by (41) will satisfy the equations and boundaryconditions of FIGS. 14a and 14b.

A certain simplification arises when in FIGS. 14a and 14b δ' andelectrode thickness are small and uniform.

The z and θ components (E_(z))_(e) and (E.sub.θ)_(e) of E_(e) are, inthe vicinity of the electrodes, the same as the values at the wall ofE_(z) and E.sub.θ in the liquid. Let (E_(z))_(l) and (E.sub.θ)_(l) bethese values at the wall in the liquid. In the same vicinity (E_(r))_(e)is however not zero (as E_(r) is on the liquid side of the wall) butgenerally the same order of magnitude as (E_(z))_(l) and (E.sub.θ)_(l).Hence the 2nd equation of (43) becomes (with respect to coordinates inFIGS. 14a and 14b ##EQU38##

Since δ' is small compared with the characteristic distance of thevariation of φ over an electrode and since φ=0 at the liquid, then onthe liquid side of an electrode (∇φ)_(n) may be replaced by φ/δ'. Sinceδ' is assumed << distances separating the other side of an electrodefrom the screen, (∇φ)_(n) is relatively small on the other side. Thus(44) becomes

    ∫∫φdθdz=0                              (46)

where the integral is carried out over the whole electrode (now a simplesurface).

Since the electrodes are thin the eddy currents in them are driven bythe components B_(r) of B and the equations for (E_(z))_(i) and(E.sub.θ)_(i) are ##EQU39##

In principle therefore we can first solve (47) for (E_(z))_(i) and(E.sub.θ)_(i) then find φ on each electrode using (45) and (46), thenfind φ in the vicinity of each electrode using (42), the first of (43)and the known value of φ on each electrode, then find E using (41) andfinally find ε using (38). However in important cases the calculation ofφ on the electrodes is sufficient to find ε.

A simple example of the application of the equations (45), (46), (47) todetermine ε is the case of a pair of strip electrodes (FIGS. 16a and16b). Here the width 2c of each electrode is assumed << the tube radiusb while the half angle αo subtended by each electrode is assumed not tobe small (i.e. α_(o) b is of the same order as b). Also the separation Δbetween the adjacent electrodes is assumed <<2c and the thickness δ ofthe insulating layer is assumed <<c. Summarising, the assumptions are##EQU40##

Under conditions (48) the equations of (45), (46), (47) are applicableand have a simple solution.

Except near the ends of the electrodes the eddy currents are mainly inthe θ direction (FIGS. 16a and 16b). (47) gives ##EQU41##

Since B_(r) is effectively independent of z across the short distance 2cthis integrates to give

    (E.sub.θ).sub.1 =-1ωB.sub.r (z+C)              (49)

where the constant C must be -(c+Δ) and (c+Δ) for the L.H. and R.H.electrode in FIG. 16(a) respectively. This value of C ensures that thetotal eddy current along each strip electrode is zero. Note that,because they are close, the eddy current patterns and (E.sub.θ)_(i) areidentical in each electrode.

Except near the ends of the electrodes equations (45) are thus ##EQU42##

In (50) we may regard ##EQU43##

Thus (E.sub.θ)_(l) can be assumed to depend on z only linearly.Equations (50) are then a consistent pair in that (l/b) δθ of the firstequals δ/δz of the second on account of the fact that ##EQU44## which isthe r component of the 2nd of (39) at the wall.

The 1st of (50) gives

    φ=(E.sub.z).sub.l (z+C)+f(θ)                     (53)

The constant C=±(c+Δ) is included in (53) for convenience. Substitutionof (53) in the 2nd of (50) and use of the 2nd of (51) gives ##EQU45##which on account of (52) reduces to ##EQU46## Hence ##EQU47##

The constant C' can be found from the requirement (46). Substituting(50) into (43) gives ##EQU48## We therefore have for φ on the electrodesthe expression ##EQU49##

If wires are connected to the electrodes as shown in FIGS. 17a and 17b.We can apply (38 and obtain a result for ε. We choose the line cde to bea straight line parallel to the z axis (FIG. 18). Now by (41) E in (38)is given by

    E=E.sub.e -∇φ

The integral ##EQU50## but the integral ##EQU51## remains the same forΔ→0. Clearly the area offered to the magnetic flux in the loop in FIG.18 can be assumed zero. Hence

    ε=φ.sub.c -φ.sub.e

Now

    φ.sub.c -φ.sub.e =φ(θ.sub.j -Δ)-φ(θ.sub.j Δ)

where φ(θ,z) is given by (56). Using (56) and (55) we thus find##EQU52## or finally ##EQU53##

The result (57) shows that in the configuration of FIG. 17 the emf ε isrelated to the value of E_(z) in the liquid directly in front of thepoint chosen to connect the ends c and e of the wires to the electrodes(first term in (57)). It is also related to the derivativeδ(E.sub.θ)_(l) /δz on z=0 over a range of angles (second term in (57)).The configuration in FIG. 17 therefore does not satisfy the conditionnecessary for self-calibration. The necessary condition could beobtained, at the cost of extra wires and electronic components bymeasuring ε at many places along the electrodes and averaging theresults (FIG. 19). This amounts to forming the integral ##EQU54##

In doing this only the first term on the R.H.S. of (57) remains and weobtain the required relation

    ε=2c(E.sub.z).sub.l

where the bar denotes mean value over the range -α_(o) <θ<α_(o).

The reason that the configuration in FIGS. 17a and 17b fails to giveεproportional to the mean value of (E_(z))_(l) over the electrodes isclearly related to the fact that ε depends on the angular position θchosen to fix the wires. This dependence on θ is in turn due to thepresence of (E.sub.θ)_(i) in the electrodes. To see this consider thesecond of (45), i.e. ##EQU55## This gives for the difference in ε forθ=0 and θ=θ_(l). ##EQU56##

Since Δis small (E.sub.θ)_(l) |z=-Δ and (E.sub.θ)_(l) |_(z) =Δ are thesame and this reduces to ##EQU57##

Thus if (E.sub.θ)_(i) was zero, ε would be independent of θ.

Now the order of magnitude of (E.sub.θ)_(i) does →0 as c→0 (see (49))but the value pf ε itself then also goes to zero at the same rate (see(57)). Therefore the problem is not resolved by further reduction of thewidth of the electrodes. We need some other way to reduce (E.sub.θ)_(i).One way is to cut slots in the electrodes FIGS. 20a and 20b so as toprevent eddy currents flowing in the θ direction but so as to retain theconnection between the various parts on the edges z=±Δ. We now studythis multi-slotted configuration.

The electrodes in FIGS. 20a and 20b are supposed to be identical tothose in FIG. 16 (except for the narrow slots ending close to the innerelectrode edges) and the conditions (48) are assumed. The width w ofeach continuous part of an electrode is assumed to be <<c.

The smallness of each continuous part of the electrodes results innegligible eddy currents. Therefore in the general equation (45)(E_(z))_(i) and (E.sub.θ)_(i) can be put equal to zero. Also (E_(z))_(l)and (E.sub.θ)_(l) are effectively constant over each continuous part.Thus by (45) we have over the nth part of the LH electrode

    φ.sub.n (θ,z)=(E.sub.z).sub.l (z-z.sub.n)+b(E.sub.θ).sub.l (θ-θ.sub.n)+C.sub.n                           (58)

where (θ_(n),z_(n)) are the coordinates of the centre of the nth partand C_(n) is a constant. Similarly on the RH electrode the potential is

    θ.sub.n (θ,z)=(E.sub.z).sub.l (z-z.sub.n)+b(E.sub.θ).sub.l (θ-θ.sub.n)+C.sub.n(59)

where (θ_(n), z_(n)) are the coordinates of the centre of the nth partof the RH electrode. Since the electrodes are close and narrow (c<<b),(E_(z))_(l) and (E.sub.θ)_(l) are the same in (59) as they are in (58).Condition (46) applied to (58) and (59) now gives ##EQU58## where thesum is carried out over all the N parts.

As shown in FIGS. 20a and 20b let the electrode wires be connected at(θ,Δ) and (θ,-Δ). The measured emf ε is independent of θ for -α_(o)<.sub.θ <α_(o) (the value of (E.sub.θ)_(i) now being negligible). On theother hand using (58) and (59) when the wires are on the nth parts (asin FIGS. 20aand 20b) the emf is ##EQU59## Since z_(n) =Δ+C and z_(n)=-Δ-C and since θ_(n) =θ_(n) we find

    ε.sub.n =2(E.sub.z).sub.l c+C.sub.n +C.sub.n

Since ε_(n) is independent of n we can sum this to obtain for the emf εmeasured at any point θ ##EQU60##

On account of (60) we have finally

    ε=2c(E.sub.z).sub.l                                (61)

as expected.

For completeness we note that the constants C_(n) and C_(n) can be foundfrom the requirement that the second of (45), with (E.sub.θ)_(i) =0 musthold on z=±Δ. Thus for the LH electrode (58) gives for the nth part

    φ.sub.n (θ,Δ)=-c(E.sub.z).sub.l +b(E.sub.θ).sub.l (θ-θ.sub.n)+C.sub.n                           (62)

This is satisfied within each part but it must also be satisfied overallfrom part to part. This requires ##EQU61## Substituting (62) in (63)gives the relation

    c.sub.n+1 =c.sub.n +c ((E.sub.z).sub.l)θ=θ.sub.n+1 -((E.sub.z).sub.l)θ=θ.sub.n !-ω((E.sub.θ).sub.l)θ-θ.sub.n

which together with ##EQU62## fixes the C_(n) in terms of (E_(z))_(l)and (E.sub.θ)_(l).

(61) is, of course, exactly the relation we require forself-calibration. The configuration of FIGS. 20a and 20b under theconditions stated above therefore provides an extension ofself-calibration to wide angle electrodes. Note that the effectiveseparation of the electrodes in measuring ε is 2c. It is natural now toinvestigate the possibility of extending the electrodes over a largerarea (i.e. dropping the condition 2c<<b). As the first step in thisdirection we consider a pair of strip electrodes of length 2c comparableto b.

We consider the configuration in FIGS. 21a and 21b under the conditions##EQU63## Over the electrodes we regard ##EQU64##

The argument is now similar to that for adjacent strip electrodes and wehave

    (E.sub.z).sub.l =1ωB.sub.r ·θb

so that on the electrodes ##EQU65## Hence ##EQU66## where = refers tothe LH and RH electrodes in FIGS. 21a and 21b respectively. Condition(46) gives ##EQU67## With wires connected as shown in FIGS. 21a and 21b##EQU68## Integrating by parts we obtain ##EQU69## where Δ has been putequal to zero on account of its smallness.

Unfortunately (65) does not generally give the required average of(E_(z))_(l) over the electrode length. It does do so if (E_(z))_(l) is alinear function of z (or a constant plus a series of odd powers of z)over the electrodes. Then, for the terms in z, z³ . z⁵ . . .cancellation occurs in the integrals in (65) and only the (constant)first term in the series need be considered. As a result we then get asrequired

    ε=2c(E.sub.z).sub.l

In general, however, this is not so. Removing (E_(z))_(i) (by egslotting in the θ direction) does not overcome the problem; equation(62) remains unchanged.

Returning to the configuration of FIGS. 20a and 20b we consider theproblem of finding ε under the relaxed condition (2c no longer <<b).Clearly the property of perfect averaging in θ will hold true in thiscase. A full treatment gives, as we would expect ##EQU70##

This suffers from the same defect as (65), i.e. averaging of (E_(z))_(l)in z is not properly achieved.

We consider now an electrode design which achieves correct averaging of(E_(z))_(l) over a large area. The configuration is illustrated in FIGS.22a and 22b. It consists of identical pairs of multi-slotted electrodeseach conforming to the design in FIGS. 20a and 20b under the conditions(48). For simplicity only three pairs are drawn in FIG. 22 but anynumber could be present. The LH sides of every pair are connected by thethin strip fgh and the RH sides of each pair are connected by the thinstrip ijk. It is important that the strips fgh and ijk lie close to eachother all along their length. The emf ε is measured across the ends ofwires connected to adjacent points c and e on any one of the electrodepairs. Let the electrode parts be numbered 1, 2 . . . 3N as shown in thefigure.

An expression for ε can be obtained in the same way as in (61). Thus(E_(z))_(l) and (E.sub.θ)_(l) can be considered constant over every partof each electrode pair and since (E_(z))_(i) and (E.sub.θ)_(i) arenegligible expressions (58) and (59) hold for φ on the nth part where nnow runs from 1 to 3N.

Since the area of the connecting strips fgh and ijk is small thecontribution by the strips to the integral in (46) is negligible and wemust have ##EQU71## Note also that

    ε=φ.sub.c -φ.sub.e

Since (E.sub.θ)_(i) =0 over the inner edge of each electrode and since(E.sub.θ)_(l) does not change over the short distance Δ, it follows thatε is the same regardless of where along the inner edges the adjacentpoints c and e are taken. This argument can be extended by allowing thepoints c and e to move (always in close proximity to each other) alongthe connecting strips fgh and ijk to another electrode pair. Summing upover every part we thus have ##EQU72## and by (67)

    ε=2c(E.sub.z).sub.l

where the bar now denotes the mean value of (E_(z))_(l) over the wholearea of the electrode.

The configuration in FIG. 22 under the conditions (48) thereforegeneralises self-calibration to large area electrodes. Note that becauseof the property of correct averaging of (E_(z))_(l) with stripelectrodes long in the flow direction when (E_(z))_(l) is a linearfunction of z a given area can be covered with fewer electrode pairsthan would otherwise be necessary.

Perhaps the most important problem in the practical realisation ofself-calibration is the effect of unwanted flux linkage. This arisesmainly in connection with flux in the space between the electrode andthe liquid.

We have seen that under certain circumstances adjacent contactlesselectrodes can be used to measure the mean value of E_(z) over theircombined area. When the spacing δ, between the electrode and the liquidis infintely small the E_(z) field sensed is (E_(z))_(l) (i.e. the zcomponent of the eddy current E field in the liquid) as required.However in practice δ, has probably to be about 2 mm from liner wear andliner strength requirements. From the θ component of the relation

    ∇×E=1ωB

we infer that the difference between the E_(z) field sensed at r=b+δ andthe (E_(z))_(l) field at r=b is

    δE.sub.z =1ωB.sub.74 δ'

On the other hand

    (E.sub.z).sub.l =1ωB.sub.c b

where B_(c) is the flux density at the flow tube centre. Hence thefractional error in the E_(z) field measurement is ##EQU73##

Generally B.sub.θ is of the same order of magnitude as B_(c) so, forexample, with o'=2 mm and b=50 mm errors of the order of 4% are likely.

This flux linkage error in E_(z) measurement causes, of course, afractional error of the same magnitude in the predicted sensitivity S.We note, however, that this error can be greatly reduced by means ofsheets of permalloy (or some other highly permeable material) placedbehind the electrodes (FIG. 23). This has the effect of reducing B.sub.θrelative to B_(c). Also unwanted flux linkage from any cause can belargely removed by zeroing the sensitivity measurement in thetransmitter while the inner wall of the flow tube is covered with alaminated and non-magnetic metal sheet. The laminated sheet must consistof strips of conductor parallel to the z axis and insulated from eachother. This makes E_(z) (but not E.sub.θ) negligible on the inner wall.

A PREFERRED EMBODIMENT OF THE INVENTION

An example of the invention wherein small electrodes are used will nowbe described with reference to FIGS. 26, 3 and 24. The flowmetersensitivity S (in FIG. 26) is given by ##EQU74## where E₁ and E₂ are thez components of E in the liquid at the positions occupied by theelectrodes (which are assumed to be small). If E₁ and E₂ can be measuredS can be deduced from (68).

Measurement of E₁ and E₂ can be accomplished using two pairs a, a' andb, b' of closely spaced point electrodes (FIG. 3). The straight line aa'and bb' joining a to a' or b to b' is parallel to the z axis and the midpoints of aa' and bb' lie in the positions normally occupied by theelectrodes that sense the flow induced potential. The distance δ betweena and a' or b and b' is small compared with the tube radius.

The emfs ε₁ and ε₂ across the remote ends of wires connected to eachpair of electrodes are proportional to E₁ and E₂ respectively providedflux linkage with each pair of wires is avoided. Neglecting E₁ in theelectrodes and wires and choosing the contour C so that it passes from ato a' via the shortest path (i.e. parallel to the z axis (see FIG. 24))we have by Ampere's circuital law

    ε.sub.1 =-E.sub.1 δ

Similarly ε₂ =-E₂ δ so that ##EQU75##

This is the relation between flowmeter sensitivity and the measured emfsε₁ and ε₂.

The relation (69) is independent of the magnetic field and thereforeremains true if the magnetic field changes for any reason (in magnitudeor in distribution) from the design magnetic field. Continuousmeasurement of ε₁ and ε₂ therefore allows continuous readjustment of thevalue of sensitivity used to convert the flow signal into a reading offlow rate. ε₁ and ε₂ are virtually independent of flow since the mainflow induced potential gradients are generated in the x direction not inthe z direction. Also the phase of ε₁ and ε₂ will in practice be almost90° removed from that of any flow signal that might be present across aand a' or b and b'. This follows because of the presence of i in therelation between E_(z) and U ##EQU76## as in equation (1)

Phase sensitive detection thus has the effect of removing any remainingflow signal across a and a' or b and b'.

It is therefore possible to make a direct measurement of sensitivitywhether or not flow is occurring in the meter. This finds application inthe calibration of newly manufactured flowmeters using a probe for E_(z)measurement in water filled meters (without the need for flow) or in theself-calibration of flowmeters working in the field.

An important advantage for short flowmeters of this method ofsensitivity measurement is that it automatically corrects for thevariation in the magnetic field due to the magnetic properties of pipesadjoining the flowmeter. It also compensates for the shorting effect offlow induced potential of conducting pipes adjoining the flowmeter. Thiscan be shown as follows.

The method of sensitivity measurement depends on the validity of therelation between flow induced potential U and the z component E_(z) ofthe eddy current E-field in the liquid. This relation is exact only in aflowmeter with infinitely long insulated tube. If the tube is insulatedfor |z|<d but conducting for |z|>d as is the case when conducting pipesadjoin the meter, the relation no longer holds generally.

The equations for E_(z) are now ##EQU77## where E_(z) ' is the electricfield component in the conducting pipes and τ is the contact impedance.

The equations for flow induced potential U for flat profile flow whenconducting pipes are present for |z|>d are ##EQU78##

Comparing (71) with (72) we see that (70) remains true so long as theright hand side of the third of (71) is zero. This is the case when τ isa function only of θ and E'_(z) is zero. These conditions may beapproximately satisfied in practice in that (a) the absolute (mean)value of τ may be more significant (for changes in sensitivity) than itsvariation with θ or z and (b) E'_(z) being due to the fringe magneticfield, may be considerably smaller than E_(z) which, arises from themain flux. A more detailed study of the relative magnitudes of E'_(z)and E_(z) confirms (b) above (see the explanation relating to tables 3,4 and 5).

In the case of a meter with small electrodes an important relation hasbeen derived above between the flow induced potential and the zcomponent of the eddy current E-field in the liquid. This relation showsa new method for measuring or monitoring flowmeter sensitivity. Therelation can be easily generalised to the case of any periodicexcitation of the magnet. Since (70) holds for any frequency ω we have,for any periodic excitation, the relation ##EQU79## from which itfollows that sensitivity is given in terms of a time integral of thevalues E₁ and E₂ of E_(z) at electrodes 1 and 2 in FIG. 26 which can bemeasured in the way described above. Sensitivity measurement based oneddy current E-field sensing is therefore possible with any kind ofperiodic magnet excitation and is not restricted to the sinewave case.It provides a method for self-calibration of electromagnetic flowmetersin the field providing automatic compensation for any variation in themagnetic field and partial compensation for changes in contact impedanceof pipes adjoining the flowmeter.

A further development of the invention is shown in FIG. 25 and consistsof an 8-electrode configuration which reduces velocity profile errors(independently of self-calibration) while retaining the self-calibrationfunction. In this case the flow signal ΔU on which the flow measurementis based is ##EQU80## and the sensitivity is now ##EQU81##

FIG. 27 is a block diagram of an example of flowmeter electronics thatcan be used with the self calibration arrangements described above.Electrodes 1, 2, 3 and 4 of the flow tube 5 (similar to the one in FIG.3) are connected to signal adding devices 6 and 8 and to signaldifferencing devices 8 and 9. The outputs of these are subtracted bydifferencing devices 10 and 11 giving signals (U₄ -U₃)-(U₂ -U₁) and (U₁+U₂)-(U₃ +U₄) respectively. These are sent to phase-sensitive detectors12 and 13 the outputs of which are proportional to |R⊥((U₄ -U₃)-(U₂-U₁))| and |R((U₁ +U₂)-(U₃ +U₄))| respectively. Where R and R⊥ have beendefined previously in connection with equations (2) and (3). The phasereference for 12 and 13 is derived from the current I used to power theelectromagnet 14 and generated by 15, the device 16 serving to convertthe alternating current to an alternating voltage. Phase-sensitivedetectors 12 and 13 differ in that the output of 12 is a dc voltageproportional to the component of the input one right angle (90°) inadvance of the reference phase while the output of 13 is a dc voltageproportional to the component of the input in phase with the reference.The dc signals from 12 and 13 are divided in 17 and then multiplied (in18) by the output of a frequency to voltage converter 19. The output of18 is thus proportional to ##EQU82## which by (4) and (5) isproportional to ΔU/S or to v (the mean velocity of the flow). Finally ascaling device 20 provides the signal for an indicator 21 of flow rate.The scaling factor of the device 20 being determined by calibration ofthe electronic circuit, by the sensitivity of the device 21 and by theelectrode separation δ and the flow tube diameter.

CALIBRATION PROBE EMBODYING THE INVENTION

The application of the invention to eddy current probes will now bedescribed with reference to FIGS. 28(a) and 28(b). Eddy current probesare devices used for measuring the sensitivity of any conventionalelectromagnetic flowmeter 51, with small electrodes 52 and 53. Theprobes are plates 54, 55 of insulating material designed to fit over theelectrodes (FIG. 28(b)). During calibration the flowtube is mounted withits axis vertical and its lower end blanked off. It is filled withelectrolyte (e.g. tap water or salted water) and with the probes inplace and with the alternating magnetic field set up, the emfs ε₄ and ε₅are measured across each pair of wires leading to small electrodes 56,57 and 58, 59 on the liquid side of the plates. The plates are fitted sothat the flowtube electrodes are covered and so that a straight linejoining each probe electrode pair is parallel to the axis of theflowtube. This can be made easier through provision of a recess 100 onthe back side of each probe and by shaping the plates so they fit thecurvature of the internal flowtube surface. The distance δ between theprobe electrodes must be small in relation to the flowtube diameter.

It will now be readily understood how the probe electrodes play the samerole as the electrode pairs in a self-calibrating electromagneticflowmeter and how the emfs ε₄ and ε₅ can be used (by means of the sameformulae) to determine the sensitivity of the flowtube. Probes can ofcourse be used for the calibration of meters with more than twoelectrodes (such as shown in FIG. 25) in a similar manner. The onlydifference between probes and self-calibrating meters is that the probes(and associated electronics) are used only occasionally to give spotreadings of sensitivity whereas self-calibrating meters continuouslycorrect for variations in sensitivity.

Calibration of electromagnetic flowmeters using eddy current probes ismore economical than conventional calibration in flow rigs especially inthe case of large size flowtubes.

Eddy current probe calibration gives only the flat profile sensitivitybut from the known approximate shape of the magnetic field, or, from theknown sensitivity verses Reynolds number characteristic, sensitivity atany Reynolds number can be inferred by calculation.

In the construction of the probes care should be taken to ensure thatprobe electrodes are as close to the tube wall as possible (probe platesas thin as possible). Also the wire 110 from electrode 56 and passingelectrode 57 should be as close as possible to the straight line joiningthe electrode centres on the surface of the plate to help reduceunwanted flux linkage as in self-calibrating flowmeters.

A glossary of the scientific symbols used in this specification is asfollows:

ROMAN

A, A magnetic potential (vector and scalar)

A_(m) (β) fourier transform of F with respect to θ and z

b tube radius

B, B magnetic flux density (vector and scalar

B_(o) design magnetic flux density

B deviation from design field B_(o)

B_(c) magnetic flux density at flowtube centre

c half-width (in z direction) of contactless electrode

C, C', C_(n), C_(n), C± constants used in contactless electrode analysis

d distance along tube axis from centre of flowtube to edge of adjoiningpipes

D (=εE) displacement current density

E, E electric field (vector and scalar)

E' electric field in (parts of the magnet) including adjoining pipes

E₁ flow induced electric field

E_(e) Eddy current electric field in insulating material in whichcontactless electrodes are normally situated but which are replaced bysimilar insulating material

E_(i) eddy current electric field inside contactless electrodes

(Eξ)_(l), (E.sub.θ)_(l) values of E.sub.ξ and E.sub.θ at the wall in theliquid

f radius of idealised magnet core

f_(o) frequency

f(θ) function used in contactless electrode analysis

F magnetic potential (B=∇F)

h half-length of coils in idealised magnet

H magnetic field strength

i √-1

I_(m) (β) modified bessel function of order m

j virtual current

j_(o) design virtual current

j' deviation from the design virtual current

j₁ flow induced current density

lτ characteristic distance over which τ varies appreciably

R(X) component of X in the reference phase direction

R⊥(X) component of X one right angle in advance of the reference phase

R₁ characteristic ratio of normal to tangential components of magneticflux density

R₂ ratio of maximum values of the z components of electric fields inadjoining pipes and in the liquid near the ends of the flowtube

S flat profile flowtube sensitivity (=ΔU/v)

t time

t₁, t₂ times at which flow signals are measured in general periodicexcitation

U flow induced electric potential

U₁, U₂ electric potentials at ends of electrode wires

U_(g) value of U on the liner wall in contactless electrode analysis

v flat profile fluid velocity

v estimated mean velocity using self-calibration

v_(o) flat velocity profile distribution

v', v' deviation from flat profile distribution

v velocity vector

V volume of integration

w width (in θ direction) of contactless electrode or of a part thereof

GREEK

α inverse of characteristic distance of variation of magnetic field inboundary condition studies

α₁, α₂ coil angles in idealised magnet

α_(o) half angle subtended by contactless electrode β dummy variable inFourier transform

δ actual of effective spacing of electrodes in an electrode pair

δs skin depth

δ' small spacing between contactless electrode and liquid

Δ half-separation between contactless electrodes

ε permittivity

μ permeability

μ_(o) permeability of free space

σ conductivity

Σ area of integration

τ contact surface impedance

φ electric potential in eddy current studies

Φ total magnetic flux through a circuit

ω angular frequency (=2πf_(o))

Ω repetition angular frequency

MISCELLANEOUS

x, y, z cartesian coordinates

r, θz polar coordinates

x_(x), x_(y), x_(z) cartesian components of X

x_(r), x.sub.θ x_(z) polar components of X

(x)_(t), (x)_(n) tangential and normal components of X

dl element of length

dv element of volume

dΣ element of area

δU, δv δs changes in flow signal, and in estimated mean velocity andsensitivity respectively ΔU flow signal ε, ε₁ ε₂ emfs generated acrosselectrode pair

short for √(α² +i μδω)

I claim:
 1. A calibration probe for measuring the sensitivity of an axially elongated electromagnetic flowmeter having an inner surface defining a flow passage having an inner diameter and a pair of diametrically opposed electrodes located on the inner surface of the flowmeter, the probe comprising:two plates of insulating material, each of said plates having a size smaller than the inner diameter of the flowmeter flow passage so that the plates can be selectively inserted into and removed from the flowmeter, an electrode-receiving surface, and two electrodes fixedly coupled to the electrode-receiving surface of each of said plates; and calibration circuitry coupled to the pair of electrodes of each of said plates and adapted to provide readings of sensitivity of the flowmeter, whereby said plates can be selectively inserted into the flow passage for calibrating the flowmeter.
 2. A calibration probe for measuring the sensitivity of an axially elongated, cylindrical electromagnetic flowmeter having an inner surface defining a flow passage with an inner diameter and a pair of diametrically opposed electrodes located on the inner surface of the flowmeter, the probe comprising:two plates of insulating material, each of said plates having a size smaller than the inner diameter of the flowmeter flow passage so that the plates can be selectively inserted into and removed from the flowmeter, an electrode-receiving surface, and two electrodes fixedly coupled to the electrode-receiving surface of each of said plates, the distance separating the electrodes of any one of said plates being small relative to the inner diameter of the flowmeter flow passage; and respective conductors coupled to the electrodes of each of said plates for electrically coupling said electrodes to calibration circuitry, whereby said plates can be selectively inserted into the flow passage for calibrating the flowmeter.
 3. A calibration probe for measuring the sensitivity of an axially elongated electromagnetic flowmeter having an inner surface defining a flow passage having an inner diameter and a pair of diametrically opposed electrodes located on the inner surface of the flowmeter, the probe comprising:two plates of insulating material, each of said plates having a size smaller than the inner diameter of the flowmeter flow passage so that the plates can be selectively inserted into and removed from the flowmeter, a first surface and a second surface, and two electrodes fixedly coupled to the first surface of each of said plates, the second surface of each of said plates including a recess, located between the electrodes of each plate, for receiving a respective one of the diametrically opposed electrodes of the flowmeter; and respective conductors coupled to the electrodes of each of said plates for electrically coupling said electrodes to calibration circuitry, whereby said plates can be selectively inserted into the flow passage for calibrating the flowmeter.
 4. A calibration probe for measuring the sensitivity of an axially elongated electromagnetic flowmeter having an inner surface defining a flow passage with an inner diameter and a contoured inner surface, and a pair of diametrically opposed electrodes located on the inner surface of the flowmeter, the probe comprising:two plates of insulating material, each of said plates having a size smaller than the inner diameter of the flowmeter flow passage so that the plates can be selectively inserted into and removed from the flowmeter, a first surface and a second surface, and two electrodes fixedly coupled to the first surface of each of said plates, the second surface of each of said plates is constructed and arranged to fit the contour of the inner surface of the flowmeter in a manner characterized in that a line joining the electrodes of any one of said plates is substantially parallel to the axis of the flowmeter; and respective conductors coupled to the electrodes of each of said plates for electrically coupling said electrodes to calibration circuitry, whereby said plates can be selectively inserted into the flow passage for calibrating the flowmeter.
 5. The calibration probe of claim 4 wherein the electromagnetic flowmeter is cylindrical and the distance separating the electrodes of any one of said plates is small relative to the inner diameter of the flowmeter flow passage.
 6. The calibration probe of claim 4 wherein the second surface of each of said plates includes a recess, located between the electrodes of each plate, for receiving a respective one of the diametrically opposed electrodes of the flowmeter.
 7. An electromagnetic flowmeter assembly comprisingan axially elongated electromagnetic flowmeter having an inner surface defining a flow passage with an inner diameter, a pair of diametrically opposed electrodes located on the inner surface of the flowmeter, and a magnetic circuit for generating a magnetic reference signal across a section of the flowmeter, calibration circuitry for providing readings of sensitivity of said flowmeter, and a calibration probe for measuring the sensitivity of the flowmeter comprising two plates of insulating material, each of said plates having: a size smaller than the inner diameter of the flowmeter flow passage so that the plates can be selectively inserted into and removed from the flowmeter; an electrode-receiving surface; two electrodes fixedly coupled to the electrode-receiving surface of each of said plates; and respective conductors coupled to the electrodes of each of said plates for electrically coupling said electrodes to said calibration circuitry.
 8. The flowmeter assembly of claim 7 wherein said calibration circuitry comprisestwo subtracting devices for providing an output signal representative of the difference between received input signals, each of said subtracting devices having an input coupled to the electrodes of a respective one of said plates and an output, a sensitivity subtracting device having an input coupled to the outputs of said subtracting devices and an output for providing a signal representative of the sensitivity of said flowmeter, and a phase-sensitive detector having an input coupled to the output of said sensitivity subtracting device and an output for providing a signal proportional to the phase component of the signal received at its input, when an electrolytic fluid medium flows through said flowmeter, which is proportional to eddy currents in said flowmeter.
 9. The calibration probe of claim 7 wherein the electromagnetic flowmeter is cylindrical and has an inner diameter, and wherein the distance separating the electrodes of any one of said plates is small relative to the inner diameter of the flowmeter.
 10. The calibration probe of claim 7 wherein each of said plates includes a second surface that has a recess, located between the electrodes of each plate, for receiving a respective one of the diametrically opposed electrodes of the flowmeter.
 11. The calibration probe of claim 7 wherein each of said plates includes a second surface constructed and arranged to fit over and cover an electrode on the inner surface of the flowmeter.
 12. The calibration probe of claim 1 wherein the electromagnetic flowmeter is cylindrical and has an inner diameter, and wherein the distance separating the electrodes of any one of said plates is small relative to the inner diameter of the flowmeter.
 13. The calibration probe of claim 1 wherein the first surface of each of said plates includes a recess, located between the electrodes of each plate, for receiving a respective one of the diametrically opposed electrodes of the flowmeter.
 14. The calibration probe of claim 1 wherein each of said plates includes a second surface constructed and arranged to fit over and cover an electrode on the inner surface of the flowmeter. 